Find the horizontal or oblique asymptotes, if any. x²+3x+2R(x) =x-1

Answered: x²+3x+2 R(x) = x-1

College Algebra

10th Edition

ISBN:9781337282291

Author:Ron Larson

Publisher:Ron Larson

ChapterP: Prerequisites

SectionP.4: Factoring Polynomials

Problem 84E: The rate of change of an autocatalytic chemical reaction is kQxkx2 where Q is the amount of the...

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Question

Find the horizontal or oblique asymptotes, if any.

Expert Solution

Step 1

Given,

$R (x) = \frac{x^{2} + 3 x + 2}{x - 1}$

We know that,

If the degree of the numerator is bigger than the degree of the denominator then there is no horizontal asymptote.

Here,

The degree of the numerator is 2 and the degree of the denominator is 1.

Therefore,

The given function does not have the horizontal asymptote.

Step 2

We know that,

Oblique asymptote occurs when the degree of the denominator is less than the degree of the numerator.

If f has oblique asymptote call it y=ax+b

As we know,

$a = \lim_{x \to \infty} \frac{f (x)}{x} b = \lim_{x \to \infty} f (x) - a_{x}$

We have,

$R (x) = \frac{x^{2} + 3 x + 2}{x - 1} \Rightarrow a = \lim_{x \to \infty} \frac{x^{2} + 3 x + 2}{x - 1} \Rightarrow \lim_{x \to \infty} \frac{x^{2} (1 - \frac{3}{x} + \frac{2}{x^{2}})}{x (1 - \frac{1}{x})} \Rightarrow a = 1$

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